Public transit agencies are not known for their flashy, up-to-date technology. In many cities, you’re lucky if your diesel bus shows up on time. But this week, the Los Angeles County Metropolitan Transportation Authority is trying something new.

Starting today, riders who live near three Metro stations will be able to download an app, tap a few times, and have a car show up at their door—or, at least, within a few blocks—and take them to that station. The ride, provided by ride-hail company Via, will cost riders with the system's TAP cards $1.75, though it will be free for those who already use Metro’s low-income subsidy programs. Riders will share their car trips with between two and five others, but the agency says they shouldn’t have to wait longer than 10 minutes for a pick-up.

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If LA has its way, the one-year experiment with on-demand service will solve the devious first-mile, last-mile problem, connecting those who live just a touch too far away from stations to get there. The idea is to make it easier for a whole new group of people to use mass transit. “We’ve created an additional layer of public transportation,” says Chris Snyder, Via’s head of global expansion. “It’s complementary.”

The timing is fortuitous: American cities and their transit agencies are hankering for innovation. In Southern California, more and more people are buying their own cars, which has put a dent in transit ridership. Plus, the agencies have felt the squeeze from tech-enabled transportation services like Uber and Lyft, and have openly fretted about the effects that autonomous vehicles (whenever they arrive) might have on their service. According to research released this month by civil engineers with the University of Kentucky, heavy rail ridership decreases 1.3 percent each year after ride-hail companies enter the average metro, and bus...